HYACINTHOS 29443

[Randy Hutson]:

 

Dear friends,

 

The locus of the P-vertex conjugate of P, as P varies on the line at infinity is a curve that is the isogonal conjugate of the anticomplementary circle.  The curve passes through the vertices of ABC and X(3446), X(3447), X(9217), and X(22259) (for P = X(513), X(523), X(512) and X(524), resp.).  If ABC is acute, the curve is closed.  If ABC is obtuse, it has as asymptotes the isogonal conjugates of the anticomplements of PU(4) (the circumcircle intercepts of the anticomplementary circle).  Is this curve a cubic or higher degree curve?  I could not find a match on Bernard's site.


[Peter Moses]:

Hi Randy,

c^6 x^2 y^2+b^2 c^2 (a^2+b^2+c^2) x^2 y z+a^2 c^2 (a^2+b^2+c^2) x y^2 z+b^6 x^2 z^2+a^2 b^2 (a^2+b^2+c^2) x y z^2+a^6 y^2 z^2 = 0
passes through
ABC, tangential tiangle, X{3446,3447,9217,22259,34130}, circular points at infinity. 

The quartic is not in Bernard's CTC.

Best regards,
Peter Moses.